Composition operators on Dirichlet-type spaces
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Abstract:
The Dirichlet-type space $D^{p}\ (1 \leq p \leq 2$) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space $A^{p}$. Let $\Phi$ be an analytic self-map of the disc and define $C_{\Phi }(f) = f \circ \Phi$ for $f \in D^{p}$. The operator $C_{\Phi }: D^{p} \rightarrow D^{p}$ is bounded (respectively, compact) if and only if a related measure $\mu _{p}$ is Carleson (respectively, compact Carleson). If $C_{\Phi }$ is bounded (or compact) on $D^{p}$, then the same behavior holds on $D^{q}\ (1 \leq q < p$) and on the weighted Dirichlet space $D_{2-p}$. Compactness on $D^{p}$ implies that $C_{\Phi }$ is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space $D^{p}$. Inner functions which induce bounded composition operators on $D^{p}$ are discussed briefly.References
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Additional Information
- R. A. Hibschweiler
- Affiliation: Department of Mathematics, University of New Hampshire, Durham, New Hampshire 03824
- Email: rah2@cisunix.unh.edu
- Received by editor(s): October 16, 1998
- Received by editor(s) in revised form: February 12, 1999
- Published electronically: August 17, 2000
- Communicated by: David R. Larson
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3579-3586
- MSC (2000): Primary 47B38; Secondary 30H05
- DOI: https://doi.org/10.1090/S0002-9939-00-05886-X
- MathSciNet review: 1778280